【質問】数学用語翻訳【スレ】

すみません。
英語の文献を全く読んだことがないので、うまく訳出できないんです。
きれいな訳語（完璧でなくても）ができる方、翻訳してください。
よろしくお願いいたします。

Groups form an important tool in the study of geometricalsymmetry.
Many geometrical objects show symmetries of varying kinds and the
natural way to classify them is by means of the groups they admit.
Later we give some examples.
For a fuller discussion we refer to the books by Speiser,Weyl and
Coxeter.

We shall need a formula for the number of orbits in a set.

THEOREM: Let G be a finite group acting on a finite set S.
For each g∈G let c_g be the number of points fixed by g.
Then the number of orbits is

t=1/|G|　X_g∈G c_g

Thus t is the "average" number of points fixed by a permutation.
To prove the theorem,we count the number of pairs (x,g)∈S×G such
that xg = x in two ways: on the one hand,for each g∈G,the number
of pairs occurring is c_g;on the other hand,for each orbit,of k
points say, each point x is fixed by the elements of stabilizer,
which by the orbit formula has |G|/k elements. Thus each orbit
contributes |G| pairs in all and so P c_g=|G|t,where t is the
number of orbits.This completes the proof.